Hmm, I'm wondering if you have a rundown for me, because I'm confused by how you are referencing your indexes of LAmbertW. I absolutely deplore Lambert's W function, and it always gives me a headache, so I've successfully avoided it my entire mathematical career.
The function:
\[
y^{1/y} : \mathbb{R}_{y\ge 1} \to [1,e^{1/e}]\\
\]
There is a critical point at \(y=e\) and this critical point is of the second order (the second derivative is non-zero). So two branches spawn from here. There exists firstly the function:
\[
f(b) : (1,e^{1/e}) \to (1,e)\\
\]
And secondly the function:
\[
g(b) : (1,e^{1/e}) \to (e, \infty)\\
\]
The first function \(f\) is expressible as:
\[
f(b) = \frac{W_0(-\log(b))}{-\log(b)}
\]
For the principle branch. Since \(W_1\) is the only other real valued branch of Lambert W (correct me if I'm wrong), I was under the impression:
\[
g(b) = \frac{W_1(-\log(b))}{-\log(b)}\\
\]
Or something similar. Maybe I've made a grave error in my assumptions. I am primarily interested in the inverse of \(y^{1/y}\) which is the other branch than the typical infinite tetration branch with the Lambert W formula.
I may have gotten ahead of myself. To set things straight, I'll ask a question:
Would I just run:
To spit out the value \(4\)? I was under the impression this would work, as the different branch of Lambert W (at the critical point \(1/e\)) would just choose the different real valued branch of the inverse of \(y^{1/y}\).
The function:
\[
y^{1/y} : \mathbb{R}_{y\ge 1} \to [1,e^{1/e}]\\
\]
There is a critical point at \(y=e\) and this critical point is of the second order (the second derivative is non-zero). So two branches spawn from here. There exists firstly the function:
\[
f(b) : (1,e^{1/e}) \to (1,e)\\
\]
And secondly the function:
\[
g(b) : (1,e^{1/e}) \to (e, \infty)\\
\]
The first function \(f\) is expressible as:
\[
f(b) = \frac{W_0(-\log(b))}{-\log(b)}
\]
For the principle branch. Since \(W_1\) is the only other real valued branch of Lambert W (correct me if I'm wrong), I was under the impression:
\[
g(b) = \frac{W_1(-\log(b))}{-\log(b)}\\
\]
Or something similar. Maybe I've made a grave error in my assumptions. I am primarily interested in the inverse of \(y^{1/y}\) which is the other branch than the typical infinite tetration branch with the Lambert W formula.
I may have gotten ahead of myself. To set things straight, I'll ask a question:
Would I just run:
Code:
-LambertW(-log(sqrt(2)),1)/log(sqrt(2))To spit out the value \(4\)? I was under the impression this would work, as the different branch of Lambert W (at the critical point \(1/e\)) would just choose the different real valued branch of the inverse of \(y^{1/y}\).
